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<title>Simulations for Statistical and Thermal Physics</title>

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<h3 style="text-align:center;">Entropy and temperature</h3>

<p class="header_title">Introduction</p>

<p>A simple calculation of the entropy of two Einstein solids that can exchange energy. The plot shows the entropy 
of the two systems, S<sub>A</sub> (<font color ="blue">blue curve</font>) and S<sub>B</sub> (<font color ="green">green curve</font>), and the entropy S (<font color ="red">red curve</font>) of the composite system. N<sub>A</sub> and N<sub>B</sub> are the number of particles in each system. The total energy is given by E = E<sub>A</sub> + E<sub>B</sub>.</p>

<p>&nbsp;&nbsp;&nbsp;&nbsp;The entropy is computed by using the binomial coefficients to determine the number of states as a function of energy. In each
case the entropy is plotted versus E<sub>A</sub>, the energy of system A. The entropy of system A is given by</p>
<p class="center">
S(E<sub>A</sub>) = k ln &#937;(E<sub>A</sub>),
</p>
<p>where &#937;<sub>A</sub>(E<sub>A</sub>) is the number of microstates of system A with energy E<sub>A</sub> and k is Boltzmann's constant. We will choose units such that k = 1. The entropy of system B is given by a similar expression. The entropy of the composite system is given by</p>
<p class="center">
S(E<sub>A</sub>) = k ln [&#937;<sub>A</sub>(E<sub>A</sub>) &#937;<sub>B</sub>(E<sub>B</sub>)] = S<sub>A</sub> + S<sub>B</sub>.
</p>

<p>The total entropy of the system is given by</p>
<p class="center">
<img src="totalentropy.jpg" alt="" align="middle" >.
</p>
<p>We will see that as the total number of particles increases, the total entropy can be approximated by
</p>
<p class="center">
<img src="entropyadditive.jpg" alt="" align="middle" >,
</p><p>where <img src="eatilde.jpg" alt="" align="bottom" > is the probable value of E<sub>A</sub>; that is, the value of E<sub>A</sub> for which S(E<sub>A</sub>) is a maximum.</p> 

<center>
<applet
 code="org.opensourcephysics.davidson.applets.ApplicationApplet.class"
 archive="./stp.jar" codebase="../" align="top" height="40"
 hspace="0" vspace="0" width="150"> <param name="target"
 value="org.opensourcephysics.stp.entropy.EntropyApp"> <param name="title"
 value="Applet"> <param name="singleapp" value="true">
</applet>
</center>

<p class="header_title">Problems</p>

<ol>

<li>Choose various values of the parameters and describe the qualitative behavior of S<sub>A</sub>, S<sub>B</sub>, and S as a function of E<sub>A</sub></li>

<li>The message box gives the relative error in the total entropy when it is approximated by its maximum value as a function of E<sub>A</sub>. Describe the behavior of this error as you increase the size of the system and keep the energy per particle a constant.</li>

<li>Discuss why the total entropy has a maximum. What can you say about the slopes of S<sub>A</sub> and S<sub>A</sub> at the value of E<sub>A</sub> where S(E<sub>A</sub>) is a maximum?</li>

<li>The values of the inverse slopes dS<sub>A</sub>/dE<sub>A</sub> and dS<sub>B</sub>/dE<sub>B</sub> can be obtained by clicking your mouse on the corresponding curves. The values of the (inverse) slopes are given in the lower right corner. We know that the two systems are in equilibrium when their temperatures are equal. Discuss the relation of the slopes 
to the temperature.</li>

</ol>

<p class="header_title">Java Classes</p>

<ul>

<li>EntropyApp</li>

</ul>

<p class = "small">Updated 8 May 2008.</p>
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